On the complete relative homology and cohomology of Frobenius extensions
type: Item
instance of: scholarly article
title: On the complete relative homology and cohomology of Frobenius extensions
zbMATH Open document ID: 0838.16007
author: T. Nozawa
published in: Tsukuba Journal of Mathematics
publication date: 1996-01-31T00:00:00Z
review text: Let \(\Lambda\) be an algebra over a commutative ring \(k\) and \(\Gamma\) a subalgebra such that the ring extension \(\Lambda/\Gamma\) is a Frobenius extension. The author introduces the complete relative homology \(H_r(\Lambda,\Gamma,-)\) and cohomology \(H^r(\Lambda,\Gamma,-)\) groups for all integers \(r\) [cf. \textit{G. Hochschild}, Trans. Am. Math. Soc. 82, 246-269 (1956; Zbl 0070.269)]\ and defines a map \(\Psi^r_{\Lambda/\Gamma}:H_r(\Lambda,\Gamma,(-)^\Delta)\to H^{-r-1}(\Lambda,\Gamma,-)\), where \(\Delta\) is the Nakayama automorphism. Then necessary and sufficient conditions under which \(\Psi^r_{\Lambda/\Gamma}\) is an isomorphism are presented. In particular, if extensions are defined by a finite group \(G\) and a subgroup \(K\) a generalization of the well-known duality for the Tate cohomology is shown.
reviewed by: Marek Golasiński
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 18G25
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 828338
zbMATH Keywords: complete cohomology
zbMATH Keywords: Hall subgroups
zbMATH Keywords: Frobenius extensions
zbMATH Keywords: complete relative homology
zbMATH Keywords: Nakayama automorphism
zbMATH Keywords: duality
zbMATH Keywords: Tate cohomology
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496162799
OpenAlex ID: W338046269
DOI: 10.21099/TKBJM/1496162799
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description: scientific article; zbMATH DE number 828338